Uniformly global observables for 1D maps with an indifferent fixed point

TL;DR

This paper investigates global-local mixing properties for 1D maps with an indifferent fixed point, introducing uniformly global observables and proving mixing for broad classes of such maps.

Contribution

It introduces the concept of uniformly global observables and proves global-local mixing for general classes of 1D maps with an indifferent fixed point.

Findings

Proved global-local mixing for uniformly global observables.

Extended results to broad classes of expanding maps with indifferent fixed points.

Established new techniques for analyzing mixing in infinite measure systems.

Abstract

We study the property of global-local mixing for full-branched expanding maps of either the half-line or the interval, with one indifferent fixed point. Global-local mixing expresses the decorrelation of global vs local observables w.r.t. to an infinite measure $\mu$. Global observables are essentially bounded functions admitting an infinite-volume average, i.e., a limit for the average of the function over bigger and bigger intervals; local observables are integrable functions (both notions are relative to $\mu$). Of course, the definition of global observable depends on the exact definition of infinite-volume average. The first choice for it would be to consider averages over the entire space minus a neighborhood of the indifferent fixed point (a.k.a. the "point at infinity"), in the limit where such neighborhood vanishes. This is the choice that was made in previous papers on the subject. The classes of systems for which global-local mixing was proved, with this natural choice of global observables, are ample but not really general. In this paper we consider uniformly global observables, i.e., $L^\infty$ functions whose averages over any interval $V$ converges to a limit, uniformly as $\mu(V) \to \infty$. Uniformly global observables form quite an extensive subclass of all global observables. We prove global-local mixing in the sense of uniformly global observables, for two truly general classes of expanding maps with one indifferent fixed point, respectively on $\mathbb{R}_0^+$ and on $(0,1]$. The technical core of the proofs is rather different from previous work.